Gauge Forces and Charge Quantization from Spectral Geometry in the Dynamical Fourier Field Framework polar form

This is the new upgraded polar-form DFF gauge paper. (The older complex-field version can be found at:https://doi.org/10.5281/zenodo.20348521) The Polar form adds new insights and shows how the universe can be explained by amplitude and phase dynamics. Abstract: The Standard Model of particle physics, despite its empirical success, relies on 19 freeparameters that must be measured rather than derived. This paper presents a reconceptu-alization of gauge theory through the Dynamical Fourier Field (DFF) framework, in whichelectromagnetic and non-Abelian gauge structures emerge from spectral geometry ratherthan being postulated as fundamental symmetries. Throughout this paper the coherencefield is written natively in polar form, Φ(k, s) = A(k, s)eiθ(k,s), and every derivation isre-checked against that explicit decomposition rather than the complex field alone.We demonstrate that gauge symmetry arises as an intrinsic redundancy of phase de-scription in a complex coherence field defined over a pre-geometric spectral manifold K.The U(1) electromagnetic structure emerges from global phase invariance, while local phasevariation necessitates a geometric connection field that projects onto the electromagneticgauge potential. Maxwell’s equations follow as geometric consequences of curvature underexplicit projection hypotheses (Lorentzian stability, local invertibility of the embedding, andthe slowly-varying window approximation) rather than independent dynamical postulates.Electric charge is shown to arise from topological winding invariants of the spectral phase,providing a geometric explanation for charge quantization without additional assumptions.The framework naturally extends to non-Abelian gauge theories, where SU(2) and SU(3)structures arise from multi-component coherence redundancy — now shown to follow fromthe same amplitude/angular decomposition as the U(1) case, Φ = ρnˆ — though the dy-namical selection principle fixing the internal dimension N to the values 1, 2, 3 observed innature remains an open problem.Gauge bosons are reinterpreted as curvature excitations; masses are derived from co-herence locking, with an explicit mass matrix vanishing exactly on unbroken generatorsand positive on broken ones; and coupling constants are shown to depend on projectionscale, with the window-integral contribution now computed exactly and the remaining de-pendence left open. Fermion mass hierarchies follow from spectral eigenvalues (establishedin Paper II), and spin statistics are derived from the non-trivial holonomy of the Spin(4)bundle over K, topologically independent of the U(1) charge winding.This work provides a single canonical definition of the projection operator PX,s usedconsistently throughout, a notation table, and a status-of-results table clearly distinguishingwhat is proved, what is derived under assumptions, and what is proposed as a conjecturalextension requiring future computation. This paper also identifies the projection of thespectral Dirac equation to its emergent-spacetime form as an open derivation rather than acompleted one; both are documented explicitly in the status table.